I get the impression from some threads that the station might be new to TtR Europe.

Any rate, I have a few questions about how these work:

1. Once you've built a station to connect to another player's routes, can you branch off those same routes (either once or multiple times) or do you need a station?

For example: Player A build a line from Danzig to Smolensk. Player B connects to those lines with a station at Danzig. Can player B then build a line to Moscow and follow it from Danzig to Smolensk, or does Player B need to build another station at Moscow?

2. If you connect to a players line, and they branch after the city, can you follow both branchings.

For example: Player A has a line that goes from Danzig to Warsaw, and them both eastward and westward. Can Player B, with a station on follow both of those branchings, or only one?

3. How does using another player's line work, in terms of itineraries? Specifically:

A. If you connect to another player's line, can you count an itinerary that runs entirely on that player's line?

For example: Player A has a line from Vienna to Athens via Zagreb, and player B builds a station in Vienna. Can Player B use this station to count a ticket that runs from Zagreb to Athens?

B. What about if you put a station on one end of the ticket with no connecting track?

For example: Player C puts a train in Athens, connecting to Player A's Athens Vienna line. Can Player C count an Athens-Zagreb ticket?

The answers may seem obvious, but I've been playing this with my family we've had arguments about the first two, and while it didn't affect the scoring this time, my father made claims about point 3 which seem contradictory to me as well.

I get the impression from some threads that the station might be new to TtR Europe.

They are indeed new to TTR Europe, and unique to that version of the board game.

Before jumping into your questions, there might be a small clarification needed. You say several times that a station connects to another player's "Routes" but that's not quite what a station does. A station only lets you use one (1) set of track from an opponent to help you on your way. This means that you can only 'rent' the track from one city to the next city...then if you want to borrow another section, you will have to play another station. This might help answer a few of your questions before even getting into specifics.

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Any rate, I have a few questions about how these work:

1. Once you've built a station to connect to another player's routes, can you branch off those same routes (either once or multiple times) or do you need a station?

For example: Player A build a line from Danzig to Smolensk. Player B connects to those lines with a station at Danzig. Can player B then build a line to Moscow and follow it from Danzig to Smolensk, or does Player B need to build another station at Moscow?

I don't have the map in front of me, but I'll do my best. If you are building your route and suddenly your brother claims a track between two cities you needed, you can use a station to borrow his track. At any point in the game, you can play the necessary cards and set a station on either city that his track touches. Then you can continue building your own track on the other side of his little interruption. If he took two sections that you needed, you will have to use two of your stations, but then you can continue on the far side. If his track touches the city you needed to get to, your station will allow you to use his track and the ticket will be complete, as long as your own track is touching the other city on your ticket.

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2. If you connect to a players line, and they branch after the city, can you follow both branchings.

For example: Player A has a line that goes from Danzig to Warsaw, and them both eastward and westward. Can Player B, with a station on follow both of those branchings, or only one?

No. Your station will only let you pick one track to follow. You must play a second station to use another line.

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3. How does using another player's line work, in terms of itineraries? Specifically:

A. If you connect to another player's line, can you count an itinerary that runs entirely on that player's line?

For example: Player A has a line from Vienna to Athens via Zagreb, and player B builds a station in Vienna. Can Player B use this station to count a ticket that runs from Zagreb to Athens?

B. What about if you put a station on one end of the ticket with no connecting track?

For example: Player C puts a train in Athens, connecting to Player A's Athens Vienna line. Can Player C count an Athens-Zagreb ticket?

You only get to use one section of track (from one city to the next city). Not the whole line. If you put a station on a spot where no track is touching it, you're wasting your station because it won't do you any good.

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The answers may seem obvious, but I've been playing this with my family we've had arguments about the first two, and while it didn't affect the scoring this time, my father made claims about point 3 which seem contradictory to me as well.

Thanks for your help,
~CMS

Let us know if things are more clear now. If not, I can try again or someone else might have a better way of explaining it. Just remember that the station only borrows one small section of track from an opponent and you should be fine!